ERRATA FOR THETA FUNCTIONS AND KNOTS

(last update: May 18, 2016)

- Page 4 line 3 from the bottom, the formula in this last display should end with \(\theta (z;\tau)\), the index \(\tau\) which shows up as an exponent of the function \(\theta\) should be deleted.
- Page 6 Example 1.1. It is \(w^2=z(z-1)(z-\lambda)\), no square root on line 15.
- Page 17 line 19: the weight written on this line should be \begin{eqnarray*} e^{i/h L(A)} \end{eqnarray*} There is an extra \(h\) (Planck's constant). This is because I was using the \hh for the font of \(h\) as Planck's constant and I forgot the backslash.
- On pages 22, 23, 24 I fell into the trap of mixing
the two conventions for the definition of the Poisson bracket. Here are
the corrections, depending on which convention you like:

__Case 1:__If you like \(\{f,g\}=\omega(X_f,X_g)\) then

on page 22 line 9 from the bottom (the display defining \(\{f,g\}\)) change the sign of the Poisson bracket;

on page 23 line 5 switch the order of the two functions in both Poisson brackets, thus make those be \(\{H,\xi_j\}\), \(\{H,\eta_j\}\);

on page 23 line 7 switch \(f\) and \(H\) in the Poisson bracket to \(\{H,f\}\).

__Case 2:__If you like \(\{f,g\}=-\omega(X_f,X_g)\), then

on page 22 line 11 from the bottom: the definition of the Poisson bracket should be \begin{eqnarray*} \{f,g\}=-\omega(X_f,X_g). \end{eqnarray*} on page 23 line 17 replace \(\{\eta_j,\xi_k\}=\delta_{jk}\) with \(\{\xi_j,\eta_k\}=\delta_{jk}\)

on page 24 line 12 from the bottom and line 8 from the bottom, replace \(\frac{i}{\hbar}\) by \(\frac{1}{i\hbar}\), that is make the correspondence principle be \begin{eqnarray*} \mbox{op}(\{f,g\})=\frac{1}{i\hbar}[\mbox{op}(f),\mbox{op}(g)]+O(\hbar). \end{eqnarray*} - Page 26 Proposition 2.1, the sign in front of the term \pi i hp^Tq in the exponent is negative. This should be in tone with Propositin 2.5 at page 54. The mistake appeared when applying the Baker-Campbell-Hausdorff formula: the exp(1/2[p^TP,q^TQ]) is missing an i^2, since we actually work with X=2\pi i p^TP and Y=2\pi iq^T Q, so there is an i^2 (plus a 2 \pi that turns \hbar into h but this I did correctly).
- Page 27 line 11, as explained above, the 1/2[p^TP,q^TQ] is missing an 2\pi i^2 in front.
- Page 45 line 5 from the bottom. the distribution associates a subspace of \(T_pM\otimes {\mathbb C}\), the index \(p\) is missing.
- Page 53 line 5 from the bottom, the variable of \(\psi\) should be \({\bf y}\) not \({\bf x}\) in the second equation of the display. Thus the function should be \(\psi({\bf y})\). Moreover, a factor of \(\psi({\bf y})\) is missing on the right side of the equation.
- Page 53 line 5 from the bottom, the exponent should of course be \(-(i/\hbar){\bf x}^T{\bf y}\), the minus is missing.
- Page 156 line 14, this is not really an error, but to be more precise, we should identify \((x,z)\) with \((x,c_{jk}(x)z)\).
- Page 156 line 12, we should point out that (although this is obvious) the cocycle condition \(c_{jk}c_{kl}c_{lj}=1\) is to be satisfied on \(U_j\cap U_k\cap U_l\) whenever this intersection is nonempty.
- Page 157 line 14, the equation on that line should be \(df_{jk}=\theta_j-\theta_k\). As published, it is missing a \(d\).
- Page 159 line 13 (this is the 3rd display), the second term in the display: \(-2\pi N\frac{1}{\hbar}{\bf y}^Td{\bf x}\) should be \(-2\pi \frac{1}{h}{\bf y}^Td {\bf x}\).
- Page 161, at the bottom: I am afraid that my explanation of the metaliniar correction is lousy, and I really like the one given in Śniatycki's book. The metalinear correction is unnecessary in the constructions from the book because the frame line bundle is trivial, however in most other situations it is necessary (including all Chern-Simons theories with non-commutative gauge group).
- Page 162, line 9 from the bottom, the equation should end with \(M_{x_j}\) and not \(M_{y_j}\).
- Page 372 condition (4) in the statement of the theorem should have, in the display, a \(Z\) in front of the parenthesis. The partition function of the cylinder should be the Fourier transform.